RESEARCH PAPERS

Main features of the formation of vortex street from free shear layers emanating from two-dimensional bluff bodies placed in uniform shear flow which is a model of a laminar boundary layer along a solid wall. This problem is concerned with the mechanism governing transition induced by small bluff bodies suspended in a laminar boundary layer. Calculations show that the background vorticity of shear flow promotes the rolling up of the vortex sheet of the same sign whereas it decelerates that of the vortex sheet of the opposite sign. The steady configuration of the conventional Karman vortex street is not possible in shear flow. Theoretical vortex patterns are experimentally examined by a flow-visualization technique.

Approximate expressions for the fluid forces acting on a central, rigid rod translating periodically in a finite length annular region of confined fluid are derived from the Navier-Stokes equations for a range of geometric and fluid parameters where viscous damping is significant. Based on the derived forces, lumped added mass, and linear dashpot modeling of an annular gap support typically found in nuclear reactors is employed to predict the fundamental frequency and modal damping of a single beam. Test methods and results for the same beam are presented which indicate the force expressions are applicable for small fluid motions.

The unsteady laminar incompressible boundary-layer flow near the three-dimensional asymmetric stagnation point has been studied under the assumptions that the free-stream velocity, wall temperature, and surface mass transfer vary arbitrarily with time. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. It is found that in contrast with the symmetric flow, the maximum heat transfer occurs away from the stagnation point due to the decrease in the boundary-layer thickness. The effect of the variation of the wall temperature with time on heat transfer is strong. The skin friction and heat transfer due to asymmetric flow only are comparatively less affected by the mass transfer as compared to those of symmetric flow.

The objective of this work is to provide a mechanical description of steady-state flow of Newtonian fluid in a branching network that consists of rigid vessels of different diameters. Solution of this problem is of importance for better understanding of the mechanics of blood flow within the microcirculation. The developed branching network model predicts a wide distribution of the hydrodynamic pressure and flow in the vessels of the same caliber (flow heterogeneity). The obtained results are compared with predictions of a simple series-parallel network model. It is shown that this model provides an accurate approximation to the values of the mean pressure and flow given by the branching network model.

After discussing the constitutive relations of inorganic glasses and their strong dependence upon temperature history, we construct a numerical algorithm in order to calculate transient and residual stresses during manufacture. The method is applied to calculating stresses generated during the annealing process of bottles.

Equations for two-phase flow are used to analyze the one-dimensional sedimentation of solid particles in a stationary container of liquid. A derivation of the equations of motion is presented which is based upon Hamilton’s extended variational principle. The resulting equations contain diffusivity terms, which are linear in the gradient of the particle concentration. It is shown that the solution of the equations for steady sedimentation is stable under small perturbations. Finally, finite-difference solutions of the equations are compared to the data of Whelan, Huang, and Copley for blood sedimentation.

Constitutive equations of elastoplastic materials with an elastic-plastic transition observed in the loading state after a first yield are presented by introducing a new parameter denoting the ratio of the size of a loading surface in the transitional state to that of a yield surface in the classical idealization which ignores the transitional state. These equations involve a reasonably simplified rule for the kinematic hardening. They would describe reasonably not only the hardening behavior but especially the softening behavior which requires our careful consideration about the elastic-plastic transition. From these equations, moreover, we derive plastic constitutive equations specifically of metals and granular media which exhibit very different plastic behaviors. Besides, brief discussions are provided concerning the existing constitutive equations describing the elastic-plastic transition.

It is well known that in a well-defined load-controlled boundary-value problem for an elastic/perfectly-plastic structure the displacements are unique if the structure is everywhere elastic, and they are not unique at the yield-point load when the structure becomes a mechanism. The present paper is concerned with the range of contained plastic deformation between these two extremes. Several examples are given in which more than one displacement field exists for loads less than the yield-point load. The significance of this phenomenon is commented on from a physical, mathematical, and computational point of view.

This paper examines the stress concentration, the yielding process, and the growth of the elastic-plastic boundary as a function of applied torque in tubular specimens with a short thin-walled section. Although the analysis is entirely quasi-static, it can, under the proper circumstances, be applied to the deformation of short specimens as generally used for dynamic testing in the torsional Kolsky bar. In the analysis, the governing equations for both elastic and elastic-plastic analyses are presented, the latter taking into account work hardening. Numerical solutions of these equations employ the finite-element method. The elastic stress distribution in the specimen and the elastic-plastic enclaves are presented for various loading stages.

A method for solving elastic-perfectly plastic problems is presented, such that the mixed elliptic-hyperbolic field may be modeled properly. The method is applied to torsion of a rectangular bar which, although coarsely meshed, exhibits features observed experimentally. In particular, initiation and growth of behavior akin to shear banding is captured. A torque-twist curve is obtained well beyond the peak torque.

A numerical scheme for time-dependent inelastic analysis of transverse deflection of plates of arbitrary shape by the boundary element method is presented in this paper. The governing differential equation is the inhomogeneous biharmonic equation for the rate of small transverse deflection. This complicated boundary-value problem for an arbitrarily shaped plate is solved by using a novel combination of the boundary element method and finite-element methodology. The number of unknowns, however, depends upon the boundary discretization and is therefore less than in a finite-element model. A combined creep-plasticity constitutive theory with state variables is used to model material behavior. The computer code developed can solve problems for an arbitrarily shaped plate with clamped or simply supported boundary conditions and an arbitrary loading history. Some illustrative numerical results for clamped and simply supported rectangular and triangular plates, under various loading histories, are presented and discussed.

Yield and fracture criteria for real materials are to a varying degree affected by a state of hydrostatic stress. Some materials, after certain deformation history, exhibit different yield point when the direction of the stress is reversed, a behavior known as the Bauschinger effect. These physical phenomena are not represented by the von Mises criterion. Based on a convexity theorem of matrices, a generalization of the von Mises criterion is presented. The new criterion satisfies the convexity requirement of plasticity theory and, with two scalar functions of deformation history α and β, produces a class of hardening behavior. The current values of α and β account for the effect of hydrostatic stress and an aspect of the Bauschinger effect on yield and fracture. The generalized criterion reduces to the form of the von Mises criterion as a special case.

Known yield functions have been constructed in the three-dimensional space of principal stresses. Their convexity in the six-dimensional space of the stress components is only conjectured. Mathematical theorems of convexity are known for functions of Hermitian matrices but have not been applied to yield functions. In this paper, Drucker’s hypothesis is properly restated, leading to convexity requirement for yield functions of elastoplastic materials. Then a special version of a known convexity theorem is presented. The theorem can be applied to construct yield functions for isotropic materials. Examples of such applications are extended to some known yield functions and other theoretically acceptable new ones.

Unstable deformations of an elastic or elastoplastic layer on an elastic or elastoplastic half space, are studied under compressive forces. Various combinations of material properties are considered, e.g., an elastic layer on an elastoplastic half space, elastoplastic layer on an elastic half space, etc. Both the flow and the total deformation plasticity models are used and the corresponding results compared. The results seem to have relevance to the problem of folding of geological formations and crustal buckling under tectonic stresses.

An Airy stress function has been used to formulate the internal stresses in an isotropic body in terms of the distribution of inelastic strains under the conditions of plane stress and plane strain. Formulation has been first obtained for the stresses in closed forms for an infinite plane containing a rectangular zone with uniform inelastic strains. Using method of Fourier transforms, analytic solutions have been obtained for the stresses due to the inclusion in a half plane with free, fixed and rigid, frictionless boundaries as well as in a layer lying on a rigid, frictionless foundation. Elastic strain energy has been formulated in closed form for the rectangular domain with uniform inelastic strain in an infinite plane and a half plane.

The present paper concerns a finite thermoelastostatic analysis considering temperature-dependent mechanical and thermal properties of material. The Lamé elastic parameters, thermal expansion coefficient, and thermal conductivity are expressed by linear functions of temperature. Higher-order terms in terms of temperature and strain invariants are introduced into the free energy expression. Lagrange formulation is employed for derivation of the governing field equations. A hollow sphere under a spherically symmetric stationary thermal field is analyzed numerically. The influence of the temperature-dependent variations of the material properties on stress state is investigated and found to be insignificant over a narrow temperature range where the material remains elastic.

It is shown that the simple Palmgren-Miner linear cumulative damage rule is a special case of a general cumulative damage theory previously established. Predictions of lifetimes for families of multistage loadings according to the Palmgren-Miner rule and the general cumulative damage theory are compared with the aim of arriving at qualitative guidelines for applicability of the Palmgren-Miner rule in cyclic loading programs.

Three-dimensional failure criteria of unidirectional fiber composites are established in terms of quadratic stress polynomials which are expressed in terms of the transversely isotropic invariants of the applied average stress state. Four distinct failure modes—tensile and compressive fiber and matrix modes—are modeled separately, resulting in a piecewise smooth failure surface.

A simple and convenient method of analysis for studying two-dimensional mixed-mode crack problems is presented. The analysis is formulated on the basis of conservation laws of elasticity and of fundamental relationships in fracture mechanics. The problem is reduced to the determination of mixed-mode stress-intensity factor solutions in terms of conservation integrals involving known auxiliary solutions. One of the salient features of the present analysis is that the stress-intensity solutions can be determined directly by using information extracted in the far field. Several examples with solutions available in the literature are solved to examine the accuracy and other characteristics of the current approach. This method is demonstrated to be superior in its numerical simplicity and computational efficiency to other approaches. Solutions of more complicated and practical engineering fracture problems dealing with the crack emanating from a circular hole are presented also to illustrate the capacity of this method.

Path-independent integrals about crack tips may be used to estimate stress-intensity factors at crack tips in plane and antiplane elasticity problems. In this paper a new class of such integrals is established by using complex stress functions and the trivial application of the Cauchy theorem of complex analysis. Both the simple Westergaard complex potentials of plane and antiplane elasticity and the more general Muskhelishvili complex potentials will be used for the construction of appropriate path-independent integrals. Two applications of these integrals to the theoretical determination of stress-intensity factors at crack tips are presented. An optical method for the experimental determination of stress-intensity factors at crack tips, based on the use of appropriate complex path-independent integrals, is also proposed.

Adhesive fracture of an interdigitated or very rough interface is investigated by considering an interface crack with no-slip zones. Both the normal and the shear stresses are singular at the crack tip with the Mode II stress-intensity factor being generally smaller than that of the Mode I.

The dynamic response of a layered composite under normal and shear impact is analyzed by assuming that the composite contains an initial flaw in the matrix material. Because of the complexities that arise from the interaction of waves scattered by the crack with those reflected by the interfaces within the composite, dynamic analyses of composites with cracks have been treated only for a few simple cases. One of the objectives of the present work is to develop an effective analytical method for determining dynamic stress solutions. This will not only lead to an in-depth understanding of the failure of composites due to impact but also provide reliable solutions that can guide the development of numerical methods. The analysis method utilizes Fourier transform for the space variable and Laplace transform for the time variable. The time-dependent angle loading is separated into two parts: one being symmetric and the other skew-symmetric with reference to the crack plane. By means of superposition, the transient boundary conditions consist of applying normal and shear tractions to a crack embedded in a layered composite. One phase of the composite could represent the fiber while the other could be the matrix. Mathematically, these conditions reduce the problem to a system of dual integral equations which are solved in the Laplace transform plane for the transform of the dynamic stress-intensity factor. The time inversion is carried out numerically for various combinations of the material properties of the composite and the results are displayed graphically.

The diffraction of time harmonic antiplane shear waves by an edge crack normal to the free surface of a homogeneous half space is considered. The problem is formulated in terms of a singular integral equation with the unknown crack opening displacement as the density function. A numerical scheme is utilized to solve the integral equation at any given finite frequency. Asymptotic solutions valid at low and high frequencies are obtained. The accuracy of the numerical solution at high frequencies and of the high frequency asymptotic solution at intermediate frequencies are examined. Graphical results are presented for the crack opening displacement and the stress intensity factor as functions of frequency and the incident angle, Expressions for the far-field displacements at high and low frequencies are also presented.

Sanders’ path-independent, energy-release rate integral I is specialized to an arbitrarily loaded shallow shell containing a stress-free void, assuming linear theory. Two forms of I are given. When the void is a crack, one form reduces to integrals over circles of vanishing radius centered at the tips and is expressible in terms of bending and stretching stressintensity factors. The other form reduces to an integral along the crack. For an elastically isotropic cylindrical shell containing a longitudinal crack and subject to a uniform bending stress at large distances from the crack, the dimensionless form of I depends on Poisson’s ratio v and a dimensionless crack length λ. When λ is small the shell is nearly flat; when λ is large the shell is very thin. An asymptotic formula is obtained for I as λ → ∞. This is done by reducing the boundary-value problem to a coupled set of singular integral equations, scaling, taking the limit as λ → ∞ to obtain inner and outer integral equations, solving the outer equations analytically, and, finally, evaluating I along the crack where the outer solutions dominate. As the evaluation of I does not require an explicit solution of the inner integral equations, an apparently intractible coupled Wiener-Hopf problem is evaded.

A general partitioned transient analysis procedure is proposed, which is amenable to a unified stability analysis technique. The procedure embodies two existing implicit-explicit procedures and one existing implicit-implicit procedure. A new implicit-explicit procedure is discovered, as a special case of the general procedure, that allows degree-by-degree implicit or explicit selections of the solution vector and can be implemented within the framework of the implicit integration packages. A new element-by-element implicit-implicit procedure is also presented which satisfies program modularity requirements and enables the use of single-field implicit integration packages to solve coupled-field problems.

Some corrections have been made hitherto to explain the great discrepancy between experimental and theoretical values of the magnetoelastic buckling field of a ferromagnetic beam plate. To solve this problem, the finite-element method was applied. A magnetic field and buckling equations of the ferromagnetic beam plate finite in size were solved numerically assuming that the magnetic torque is proportional to the rotation of the plate and by using a disturbed magnetic torque deduced by Moon. Numerical and experimental results agree well with each other within 25 percent.

In this paper, a modified theory based upon Reissner’s procedure for the shear-lag effect of the sandwich panel is presented, which includes the effects of the anisotropy of the faces and the shearing rigidity of the core. In order to verify this theory, bending experiments were performed with sandwich panels composed of a soft core, stiffeners, and orthotropic faces. It was found that the effective bending rigidity calculated from this theory was lower than that derived from the classical bending theory and that the theoretical strain distribution on the faces agreed well with the experimental results.

The equations for static torsional deformation of a pretwisted beam under axial loading are obtained from the principle of virtual work. Theory appropriate for a curvilinear coordinate system is used, and warp is included in the analysis from the outset. The present formulation yields the expected result that a pretwisted beam of noncircular cross section will untwist under tensile loading. A slender-beam approximation of the present theory is offered, and the resulting torsion-extension coupling terms are similar but not identical to those in common use in analyses of rotating blades. Numerical results indicate that the effect is not negligible when the ratio of shear modulus to extension modulus is small.

Some work on the in-plane vibrations and buckling of rotating beams, clamped off the axis of rotation, is unclear as to behavior in the limit of small stiffness and small off-clamping. In this paper, asymptotic expansion formulas are developed for both small and large values of stiffness and off-clamping parameters. A composite expansion formula is also introduced as an engineering approximation to the buckling curve for all values of parameters. The present results agree quite well with an exact numerical solution and indicate that buckling can occur for arbitrarily small stiffness and off-clamping.

Hoff’s problem—that of investigation of the maximum load supported by an elastic column in a compression test—is considered in a probabilistic setting. The initial imperfections are assumed to be Gaussian random fields with given mean and autocorrelation functions, and the problem is solved by the Monte Carlo Method. The Fourier coefficients of the expansion for the initial imperfection function are simulated numerically; for each realization of the initial imperfection function, the maximum load supported by an elastic column in a compression test is found by the solution of a set of coupled nonlinear differential equations. For slightly imperfect columns, the closed solution is given in terms of Bessel and Lommel functions and turns out to compare well with the result of numerical integration. Results of the Monte Carlo solution are used in constructing the reliability function at a specified load. Reliability functions for different manufacturing processes (represented by different autocorrelation functions with equal variance) are calculated; design requirement suggests then that, other conditions being equal, the preference should be given to the manufacturing process resulting higher reliability.

A canonical perturbation method based on Hamilton-Jacobi theory, used together with Galerkin’s method, is employed to analyze the nonlinearly coupled transverse free oscillations of columns subjected to a constant end force. Integrals of motion, readily obtained from this type of analysis, are used to allow the analytical determination of the main characteristics of the resonant motion and of the region of resonance of the system.

This paper studies the controllability and observability of the system Mq̈ + Gq̇ + Kq = Bu , where M is symmetric and positive-definite, G is skew-symmetric and K is symmetric. In all cases, the output equation is y = Pq + Rq̇ . This special structure is exploited to derive relatively simple controllability and observability conditions which are shown to provide important insights on the modal behavior of the system and to furnish information on the number and positioning of sensors and actuators.

This paper is concerned with the generalization of Poincaré’s theory of index to systems of order higher than two. The basic tool used in the generalization is the concept of the degree of a map. In topology this concept has been used to discuss the index of a vector field. In this paper we shall use the degree of a map concept to present a theory of index for higher-order systems in a form which might make it more accessible to engineers for applications. The theory utilizes the notion of the index of a hypersurface with respect to a given vector field. After presenting the theory, it is applied to dynamical systems governed by ordinary differential equations and also to dynamical systems governed by point mappings. Finally, in order to show how the abstract concept of the degree of a map, hence the index of a surface, may actually be evaluated, illustrative procedures of evaluation for two kinds of hypersurfaces are discussed in detail and an example of application is given.

TECHNICAL BRIEFS

This work concerns a pendulum made up of a mass and a weightless rod attached to a pivot. The mass is constrained to move rotationally with the rod but is free to move along the rod. Horizontal vibration of the pivot causes an inertia force in the negative radial direction which prevents the mass from sliding off the rod. Stability of an equilibrium solution is determined.

The purpose of the present Note is to show how the general axisymmetric photoelastic problem can be solved using only the integrated data. Indeed, the axisymmetric problem is reduced to a quasi-plane problem in that only a fringe photograph plus a plot of the family of isoclinics is sufficient to yield a complete solution. The formulation includes the treatment of thermoelastic and residual stress problems.

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